# Issue: REAL-NUMBER-TYPE (version 3)

```Issue:        REAL-NUMBER-TYPE
Forum:	      CLEANUP
References:   Table 4-1.
Edit history: 04-JAN-89, Version 1 by Bob Cassels, Don Sakahara, Kent Pitman,
and John Aspinall
08-JAN-89, Version 2 by Bob Cassels -- incorporate
Masinter's suggestion and make REAL a CLOS class
13-JAN-89, Version 3 by Cassels and Aspinall -- incorporate Marc LeBrun's
suggestions clarifying the relationship between CL
numeric type names and mathematical names
Status:	      For Internal Discussion

Problem Description:

There is no standard type specifier symbol for the CL type
'(OR RATIONAL FLOAT).

Proposal (REAL-NUMBER-TYPE:REAL):

Make REAL be a CL data type:

p.13 "Numbers"

Add:     The NUMBER data type encompasses all of these kinds of
numbers.  For convenience, there are names for some
subclasses of numbers.  @i[Integers] and @i[ratios] are of
type RATIONAL.  @i[Rational numbers] and @[floating-point
numbers] are of type REAL.  @i[Real numbers] and @i[complex
numbers] are of type NUMBER.

Although the names of these types were chosen with the
terminology of mathematics in mind, the correspondences
are not always exact.  Integers and ratios model the
corresponding mathematical concepts directly.  Numbers
of the FLOAT type may be used to approximate real
numbers, both rational and irrational.  The REAL type
includes all Common Lisp numbers which represent
mathematical real numbers, though there are
mathematical real numbers (irrational numbers)
which do not have an exact Common Lisp representation.
Only REAL numbers may be ordered using the <, >, <=,
and >= functions.

Compatibility note:  The Fortran standard defines the term
"real datum" to mean "a processor approximation to the value
of a real number."  In practice the Fortran "basic real" type
is the floating-point data type Common Lisp calls
SINGLE-FLOAT.  The Fortran "double precision" type is
Common Lisp's DOUBLE-FLOAT.  The Pascal "real" data type is
an "implementation-defined subset of the real numbers."  In
practice this is usually a floating-point type, often what
Common Lisp calls DOUBLE-FLOAT.

A translation of an algorithm written in Fortran or Pascal
which uses "real" data usually will use some appropriate
precision of Common Lisp's FLOAT type.  Some algorithms may
gain accuracy and/or flexibility by using Common Lisp's

p.33 "Overlap, Inclusion, and Disjointness of Types":

Remove:  The types RATIONAL, FLOAT, and COMPLEX are pairwise
disjoint subtypes of NUMBER.

Rationale: It might be thought that INTEGER and RATIO ...

Rationale: It might be thought that FIXNUM and BIGNUM ...

Add:     The types RATIONAL and FLOAT are pairwise disjoint subtypes
of REAL.

The types REAL and COMPLEX are pairwise disjoint subtypes
of NUMBER.

Rationale: It might be thought that FIXNUM and BIGNUM should
form an exhaustive partition of the type INTEGER, that INTEGER
and RATIO should form an exhaustive partition of RATIONAL,
that RATIONAL and FLOAT should form an exhaustive partition of
REAL, and that REAL and COMPLEX should form an exhaustive
partition of NUMBER.  These are all purposely avoided in order
to permit compatible experimentation with extensions to the
Common Lisp number system, such as the idea of adding explicit
representations of infinity or of positive and negative infinity.

p.43 Table 4-1 "Standard Type Specifier Symbols"

p.49 "Type Specifiers that Abbreviate"

Denotes the set of real numbers between low and high.  ...
[As with RATIONAL and FLOAT.]

Make REAL a built-in CLOS class.

Proposal (REAL-NUMBER-TYPE:REALP):

Add a specific data type predicate REALP which tests for membership in
this type.  [By analogy with NUMBERP.]

Test Case:

If a programmer wishes to test for "a number between 1 and 10", the
only current CL types would be '(or (rational 1 10) (float 1 10)) or
something like '(and numberp (not complexp) (satisfies range-1-10))
with (defun range-1-10 (real) (<= 1 real 10)).  Both of these are
likely less efficient, and certainly less expressive than '(real 1 10).

Rationale:

Mathematics has a name for (OR RATIONAL FLOAT) -- it is "real".
This class is important because it is all the numbers which can be
ordered.

Throughout the "Numbers" chapter, the phrase "non-complex number" is
used.
MAX, MIN, p. 198 "The arguments may be any non-complex numbers."
CIS p. 207 "The argument ... may be any non-complex number."

Current Practice:

Probably nobody does this.

Cost to Implementors:

Some work is necessary to add this name.  But since the underlying
type already exists the amount of work should be minimal.

Cost to Users:

Since this is an upward-compatible extension, it may be ignored by
users.

Occasional inconvenience and/or inefficiency.

Benefits:

Mathematical clarity.

Ability to do CLOS method dispatch on the type.

Aesthetics:

As mentioned under "rationale," this would be a more concise way to
express a common programming idiom.

Discussion:

The name "non-complex number" is incorrect because future
implementations may wish to include numerical types which are neither
complex nor real.  [e.g. pure imaginary numbers or quaternions]

The name "scalar" is incorrect because the mathematical concept of
scalar may indeed include complex numbers.

Fortran and Pascal use the name "real" to mean what CL calls
SINGLE-FLOAT.  That should cause no significant problem, since a Lisp
program written using the type REAL will do mathematically what the
equivalent Fortran program would do.  This is because Fortran's "real"
data-type is a subtype of the CL REAL type.  The only differences
might be that the Lisp program could be less efficient and/or more
accurate.

A survey of several Fortran and Pascal books shows that the distinction
between INTEGER and REAL is that REAL numbers may have fractional
parts, while INTEGERs do not.  Later discussions explain that REALs
cover a greater range.  Much later discussions cover precision
considerations, over/underflow, etc.  So the average Fortran or Pascal
programmer should be completely comfortable with the proposed Lisp
concept of REAL.

```